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The eigenvalues of the empirical transition matrix of a Markov chain

Part of: Limit theorems Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Geoffrey Pritchard  and
David J. Scott
Geoffrey Pritchard
Affiliation:
Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand. Email address: g.pritchard@stat.auckland.ac.nz
David J. Scott
Affiliation:
Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand. Email address: d.scott@auckland.ac.nz
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Abstract

This paper investigates the probabilistic behaviour of the eigenvalue of the empirical transition matrix of a Markov chain which is of largest modulus other than 1, loosely called the second-largest eigenvalue. A central limit theorem is obtained for nonmultiple eigenvalues of the empirical transition matrix. When the Markov chain is actually a sequence of independent observations the distribution of the second-largest eigenvalue is determined and a test for independence is developed. The independence case is considered in more detail when the Markov chain has only two states, and some applications are given.

Keywords

Markov chaineigenvaluerandom matrix

MSC classification

Primary: 62M05: Markov processes
Secondary: 60F05: Central limit and other weak theorems 62M02: Markov processes
Type
Part 6. Stochastic processes
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 347 - 360
Copyright
Copyright © Applied Probability Trust 2004 

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